\chapter{第五周作业}
\section{计算偏导数}

\begin{enumerate}
    \item \(\begin{aligned}
        z=\tan\frac{x^2}y;
    \end{aligned}\)

    \item \(\begin{aligned}
        z=x^y
    \end{aligned}\)

    \item $\begin{aligned}
        u=e^{xyz}
    \end{aligned}$
    \item \(\begin{aligned}
        u=x^{yz}
    \end{aligned}\)
    \item \(\begin{aligned}
        u=\ln(x+y^2+z^3)
    \end{aligned}\)

    \item \(\begin{aligned}
    z=\arcsin{(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})}
    \end{aligned}\)
\end{enumerate}

\section{计算下列函数在指定点处的全微分}
\begin{enumerate}
    \item \(\begin{aligned}
        f(x,y)=x^2+2xy-y^2,\text{在点}(1,2)\text{处;}
    \end{aligned}\)
    \item \(\begin{aligned}
        f(x,y,z)=\ln{(x+y-z)}+e^{x+y}\sin{z},\text{在点}(1,2,1)\text{处;}
    \end{aligned}\)
    \item \(\begin{aligned}
        u=\sqrt{x_1^2+x_2^2+\cdots+x_n^2},\text{在点(}t_1,t_2,\cdots,t_n)\text{处,其中 }\quad t_1^2+t_2^2+\cdots+t_n^2>0;
        \end{aligned}\)
\end{enumerate}
\section{补充（无需写在作业里）}
证明：二元函数
\[
f(x,y)=\begin{cases}(x^2+y^2)\sin\dfrac1{x^2+y^2},&(x,y)\neq(0,0),\\\\0,&(x,y)=(0,0)\end{cases}
\]
在\((0,0)\)处可微，但是在两个偏导数\(\begin{aligned}
    \frac{\partial f}{\partial x},\frac{\partial f}{\partial y}
\end{aligned}\)在$(0,0)$处不连续，由此得出\emph{\textbf{函数可微但是各偏导数不一定连续，即各偏导数连续是可微的充分条件但不是必要条件}}。
\begin{proof}
    不难求出：
    \[
    \dfrac{\partial f}{\partial x}=\begin{cases}2x\sin\dfrac{1}{x^2+y^2}-\dfrac{2x}{x^2+y^2}\cos\dfrac{1}{x^2+y^2},&(x,y)\neq(0,0)\\0&(x,y)=0\end{cases}
    \]
    以及
    \[
    \frac{\partial f}{\partial y}=\begin{cases}2y\sin\dfrac{1}{x^2+y^2}-\dfrac{2y}{x^2+y^2}\cos\dfrac{1}{x^2+y^2},&(x,y)\neq(0,0)\\0,&(x,y)=0\end{cases},
    \]
由此可见$\dfrac{\partial f}{\partial x}$和$\dfrac{\partial f}{\partial y}$在$\left(0,0\right)$处都不连续。因为
\[
\lim_{(x,y)\to(0,0)}\frac{f(x,y)-xf_x(0,0)-yf_y(0,0)-f(0,0)}{\sqrt{x^2+y^2}}=\lim_{(x,y)\to(0,0)}\sqrt{x^2+y^2}\sin\frac{1}{x^2+y^2}=0,
\]，所以$f(x,y)$在$(0,0)$处可微.
\end{proof}

\section{参考答案}
\subsection{计算偏导数}
\begin{enumerate}
    \item \(z_x=2(x/y)\sec^2(x^2/y),z_y=-(x/y)^2\sec^2(x^2/y)\)

    \item \(z_{x}=yx^{y-1},z_{y}=x^{y}\ln x\)
    \item \(u_x=yz\mathrm{e}^{xyz},u_y=xz\mathrm{e}^{xyz},u_z=xy\mathrm{e}^{xyz}\)
    \item \(u_{x}=yzx^{yz-1},u_{y}=x^{yz}z\ln x,u_{z}=x^{yz}y\ln x\)
    \item \(u_x=1/(x+y^2+z^3),u_y=2y/(x+y^2+z^3),u_z=3z^2/(x+y^2+z^3)\)
    \item \(z_{x_i}=2x_i/\sqrt{1-(x_1^2+x_2^2+\cdots+x_n^2)^2}\)
\end{enumerate}
\subsection{计算下列函数在指定点处的全微分}
\begin{enumerate}
    \item \(6\mathrm{d}x-2\mathrm{d}y\)
    \item \((1/2+\mathrm{e}^3\sin1)\mathrm{d}x+(1/2+\mathrm{e}^3\sin1)\mathrm{d}y+(-1/2+\mathrm{e}^3\cos1)\mathrm{d}z\)
    \item \(\sum_{i=1}^n\frac{t_i\mathrm{d}x_i}{\sqrt{t_1^2+t_2^2+\cdots+t_n^2}}\)
\end{enumerate}